Posterior Summary and Visualization Utilities

This vignette demonstrates the summary and plotting utilities available for stochtree models.

Setup

Load necessary packages

library(stochtree)

import numpy as np
import matplotlib.pyplot as plt
from stochtree import BARTModel, BCFModel, plot_parameter_trace

Set a seed for reproducibility

random_seed = 1234
set.seed(random_seed)

random_seed = 1234
rng = np.random.default_rng(random_seed)

Supervised Learning

We begin with the supervised learning use case served by the bart() function.

Below we simulate a simple regression dataset.

n <- 1000
p_x <- 10
p_w <- 1
X <- matrix(runif(n * p_x), ncol = p_x)
W <- matrix(runif(n * p_w), ncol = p_w)
f_XW <- (((0 <= X[, 10]) & (0.25 > X[, 10])) *
  (-7.5 * W[, 1]) +
  ((0.25 <= X[, 10]) & (0.5 > X[, 10])) * (-2.5 * W[, 1]) +
  ((0.5 <= X[, 10]) & (0.75 > X[, 10])) * (2.5 * W[, 1]) +
  ((0.75 <= X[, 10]) & (1 > X[, 10])) * (7.5 * W[, 1]))
noise_sd <- 1
y <- f_XW + rnorm(n, 0, 1) * noise_sd

n = 1000
p_x = 10
p_w = 1
X = rng.uniform(size=(n, p_x))
W = rng.uniform(size=(n, p_w))
# R uses X[,10] (1-indexed) = Python X[:,9]
f_XW = (
    ((X[:, 9] >= 0)    & (X[:, 9] < 0.25)) * (-7.5 * W[:, 0]) +
    ((X[:, 9] >= 0.25) & (X[:, 9] < 0.5))  * (-2.5 * W[:, 0]) +
    ((X[:, 9] >= 0.5)  & (X[:, 9] < 0.75)) * ( 2.5 * W[:, 0]) +
    ((X[:, 9] >= 0.75) & (X[:, 9] < 1.0))  * ( 7.5 * W[:, 0])
)
noise_sd = 1.0
y = f_XW + rng.standard_normal(n) * noise_sd

Now we fit a simple BART model to the data.

num_gfr <- 10
num_burnin <- 0
num_mcmc <- 1000
general_params <- list(
  num_threads = 1, 
  num_chains = 3
)
bart_model <- stochtree::bart(
  X_train = X,
  y_train = y,
  leaf_basis_train = W,
  num_gfr = num_gfr,
  num_burnin = num_burnin,
  num_mcmc = num_mcmc,
  general_params = general_params
)

bart_model = BARTModel()
bart_model.sample(
    X_train=X,
    y_train=y,
    leaf_basis_train=W,
    num_gfr=10,
    num_burnin=0,
    num_mcmc=1000,
    general_params={
      "num_threads": 1, 
      "num_chains": 3
    },
)

We obtain a high level summary of the BART model by running print().

print(bart_model)

stochtree::bart() run with mean forest, global error variance model, and mean forest leaf scale model
Continuous outcome was modeled as Gaussian with a leaf regression prior with 1 bases for the mean forest
Outcome was standardized
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

print(bart_model)

BARTModel run with mean forest, global error variance model, and mean forest leaf scale model
Outcome was modeled as gaussian with a leaf regression prior with 1 bases for the mean forest
Outcome was standardized
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

For a more detailed summary (including the information above), we use the summary() function.

summary(bart_model)

stochtree::bart() run with mean forest, global error variance model, and mean forest leaf scale model
Continuous outcome was modeled as Gaussian with a leaf regression prior with 1 bases for the mean forest
Outcome was standardized
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning) 
Summary of sigma^2 posterior: 
3000 samples, mean = 0.892, standard deviation = 0.050, quantiles:
     2.5%       10%       25%       50%       75%       90%     97.5% 
0.7996598 0.8290642 0.8570033 0.8907036 0.9256984 0.9577728 0.9970341 
Summary of leaf scale posterior: 
3000 samples, mean = 0.007, standard deviation = 0.001, quantiles:
       2.5%         10%         25%         50%         75%         90% 
0.004856882 0.005390241 0.005887984 0.006497165 0.007306410 0.008073880 
      97.5% 
0.009231537 
Summary of in-sample posterior mean predictions: 
1000 observations, mean = -0.065, standard deviation = 3.284, quantiles:
      2.5%        10%        25%        50%        75%        90%      97.5% 
-6.8624792 -4.9294766 -1.8221348 -0.1219754  2.0219140  4.2918042  6.5265284

print(bart_model.summary())

BART Model Summary:
-------------------
BARTModel run with mean forest, global error variance model, and mean forest leaf scale model
Outcome was modeled as gaussian with a leaf regression prior with 1 bases for the mean forest
Outcome was standardized
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

Summary of sigma^2 posterior: 3000 samples, mean = 0.941, standard deviation = 0.047, quantiles:
    2.5%: 0.856
   10.0%: 0.880
   25.0%: 0.908
   50.0%: 0.939
   75.0%: 0.973
   90.0%: 1.005
   97.5%: 1.039
Summary of leaf scale posterior: 3000 samples, mean = 0.007, standard deviation = 0.001, quantiles:
    2.5%: 0.005
   10.0%: 0.006
   25.0%: 0.006
   50.0%: 0.007
   75.0%: 0.008
   90.0%: 0.008
   97.5%: 0.009
Summary of in-sample posterior mean predictions: 
1000 observations, mean = 0.106, standard deviation = 3.286, quantiles:
    2.5%: -6.745
   10.0%: -4.328
   25.0%: -1.909
   50.0%: 0.170
   75.0%: 2.074
   90.0%: 4.405
   97.5%: 6.614

None

We can use the plot() function to produce a traceplot of model terms like the global error scale \(\sigma^2\) or (if \(\sigma^2\) is not sampled) the first observation of cached train set predictions.

plot(bart_model)

ax = plot_parameter_trace(bart_model, term="global_error_scale")
plt.show()

For finer-grained control over which parameters to plot, we can also use the extractParameter() function to pull the posterior distribution of any valid model term (e.g., global error scale \(\sigma^2\), leaf scale \(\sigma^2_{\ell}\), in-sample mean function predictions y_hat_train) and then plot any subset or transformation of these values.

y_hat_train_samples <- extractParameter(bart_model, "y_hat_train")
obs_index <- 1
plot(
  y_hat_train_samples[obs_index, ],
  type = "l",
  main = paste0("In-Sample Predictions Traceplot, Observation ", obs_index),
  xlab = "Index",
  ylab = "Parameter Values"
)

y_hat_train_samples = bart_model.extract_parameter("y_hat_train")
obs_index = 0
fig, ax = plt.subplots()
ax.plot(y_hat_train_samples[obs_index, :])
ax.set_title(f"In-Sample Predictions Traceplot, Observation {obs_index}")
ax.set_xlabel("Index")
ax.set_ylabel("Parameter Values")
plt.show()

Causal Inference

We now run the same demo for the causal inference use case served by the bcf() function in R and the BCFModel Python class.

Below we simulate a simple dataset for a causal inference problem with binary treatment and continuous outcome.

# Generate covariates and treatment
n <- 1000
p_X = 5
X = matrix(runif(n * p_X), ncol = p_X)
pi_X = 0.25 + 0.5 * X[, 1]
Z = rbinom(n, 1, pi_X)

# Define the outcome mean functions (prognostic and treatment effects)
mu_X = pi_X * 5 + 2 * X[, 3]
tau_X = X[, 2] * 2 - 1

# Generate outcome
epsilon = rnorm(n, 0, 1)
y = mu_X + tau_X * Z + epsilon

# Generate covariates and treatment
n = 1000
p_X = 5
X = rng.uniform(size=(n, p_X))
pi_X = 0.25 + 0.5 * X[:, 0]
Z = rng.binomial(1, pi_X, n).astype(float)

# Define the outcome mean functions (prognostic and treatment effects)
mu_X = pi_X * 5 + 2 * X[:, 2]
tau_X = X[:, 1] * 2 - 1

# Generate outcome
epsilon = rng.standard_normal(n)
y = mu_X + tau_X * Z + epsilon

Now we fit a simple BCF model to the data

num_gfr <- 10
num_burnin <- 0
num_mcmc <- 1000
general_params <- list(
  num_threads = 1, 
  num_chains = 3, 
  adaptive_coding = TRUE
)
bcf_model <- stochtree::bcf(
  X_train = X,
  y_train = y,
  Z_train = Z,
  num_gfr = num_gfr,
  num_burnin = num_burnin,
  num_mcmc = num_mcmc,
  general_params = general_params
)

bcf_model = BCFModel()
bcf_model.sample(
    X_train=X,
    Z_train=Z,
    y_train=y,
    propensity_train=pi_X,
    num_gfr=10,
    num_burnin=0,
    num_mcmc=1000,
    general_params={
      "num_threads": 1, 
      "num_chains": 3, 
      "adaptive_coding": True
    },
)

We obtain a high level summary of the BCF model by running print().

print(bcf_model)

stochtree::bcf() run with prognostic forest, treatment effect forest, global error variance model, prognostic forest leaf scale model, and treatment effect intercept model
Outcome was modeled as gaussian
Treatment was binary and its effect was estimated with adaptive coding
outcome was standardized
An internal propensity model was fit using stochtree::bart() in lieu of user-provided propensity scores
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

print(bcf_model)

BCFModel run with prognostic forest, treatment effect forest, global error variance model, prognostic forest leaf scale model, and treatment effect intercept model
Outcome was modeled as gaussian
Treatment was binary and its effect was estimated with adaptive coding
Outcome was standardized
User-provided propensity scores were included in the model
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

For a more detailed summary (including the information above), we use the summary() function / method.

summary(bcf_model)

stochtree::bcf() run with prognostic forest, treatment effect forest, global error variance model, prognostic forest leaf scale model, and treatment effect intercept model
Outcome was modeled as gaussian
Treatment was binary and its effect was estimated with adaptive coding
outcome was standardized
An internal propensity model was fit using stochtree::bart() in lieu of user-provided propensity scores
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning) 
Summary of sigma^2 posterior: 
3000 samples, mean = 0.947, standard deviation = 0.044, quantiles:
     2.5%       10%       25%       50%       75%       90%     97.5% 
0.8625807 0.8915014 0.9168516 0.9458133 0.9752869 1.0048642 1.0350298 
Summary of prognostic forest leaf scale posterior: 
3000 samples, mean = 0.002, standard deviation = 0.000, quantiles:
       2.5%         10%         25%         50%         75%         90% 
0.000931787 0.001061290 0.001221171 0.001446308 0.001731303 0.002026387 
      97.5% 
0.002487405 
Summary of adaptive coding parameters: 
3000 samples, mean (control) = -0.478, mean (treated) = 1.013, standard deviation (control) = 0.378, standard deviation (treated) = 0.340
quantiles (control):
        2.5%          10%          25%          50%          75%          90% 
-1.243155421 -0.995781380 -0.722984496 -0.454813405 -0.205130109 -0.008770076 
       97.5% 
 0.179493189 
quantiles (treated):
     2.5%       10%       25%       50%       75%       90%     97.5% 
0.3791020 0.5886795 0.7753516 0.9940181 1.2321716 1.4616540 1.7391566 
Summary of treatment effect intercept (tau_0) posterior: 
3000 samples, mean = 0.198, standard deviation = 0.830, quantiles:
      2.5%        10%        25%        50%        75%        90%      97.5% 
-1.7117909 -1.1280350 -0.2202302  0.2092770  0.8450330  1.2141855  1.5190689 
Summary of in-sample posterior mean predictions: 
1000 observations, mean = 3.487, standard deviation = 0.945, quantiles:
    2.5%      10%      25%      50%      75%      90%    97.5% 
1.652175 2.280813 2.804716 3.483329 4.132789 4.713217 5.431474 
Summary of in-sample posterior mean CATEs: 
1000 observations, mean = 0.066, standard deviation = 0.523, quantiles:
       2.5%         10%         25%         50%         75%         90% 
-0.73977684 -0.56669850 -0.39984063 -0.01084696  0.56586896  0.77474896 
      97.5% 
 0.90562238

print(bcf_model.summary())

BCF Model Summary:
------------------
BCFModel run with prognostic forest, treatment effect forest, global error variance model, prognostic forest leaf scale model, and treatment effect intercept model
Outcome was modeled as gaussian
Treatment was binary and its effect was estimated with adaptive coding
Outcome was standardized
User-provided propensity scores were included in the model
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

Summary of sigma^2 posterior: 3000 samples, mean = 0.872, standard deviation = 0.042, quantiles:
    2.5%: 0.794
   10.0%: 0.817
   25.0%: 0.843
   50.0%: 0.870
   75.0%: 0.899
   90.0%: 0.926
   97.5%: 0.954
Summary of prognostic forest leaf scale posterior: 3000 samples, mean = 0.002, standard deviation = 0.000, quantiles:
    2.5%: 0.001
   10.0%: 0.001
   25.0%: 0.001
   50.0%: 0.002
   75.0%: 0.002
   90.0%: 0.002
   97.5%: 0.003
Summary of adaptive coding parameters: 
3000 samples, mean (control) = -0.574, mean (treated) = 1.159, standard deviation (control) = 0.374, standard deviation (treated) = 0.362
quantiles (control):
    2.5%: -1.380
   10.0%: -1.048
   25.0%: -0.785
   50.0%: -0.539
   75.0%: -0.325
   90.0%: -0.139
   97.5%: 0.092

quantiles (treated):
    2.5%: 0.472
   10.0%: 0.707
   25.0%: 0.912
   50.0%: 1.143
   75.0%: 1.406
   90.0%: 1.636
   97.5%: 1.875
Summary of treatment effect intercept (tau_0) posterior: 3000 samples, mean = 0.137, standard deviation = 0.436, quantiles:
    2.5%: -0.539
   10.0%: -0.406
   25.0%: -0.178
   50.0%: 0.093
   75.0%: 0.384
   90.0%: 0.815
   97.5%: 1.134
Summary of in-sample posterior mean predictions: 
1000 observations, mean = 3.481, standard deviation = 0.961, quantiles:
    2.5%: 1.844
   10.0%: 2.207
   25.0%: 2.772
   50.0%: 3.473
   75.0%: 4.158
   90.0%: 4.771
   97.5%: 5.321
Summary of in-sample posterior mean CATEs: 
1000 observations, mean = -0.028, standard deviation = 0.628, quantiles:
    2.5%: -1.085
   10.0%: -0.918
   25.0%: -0.582
   50.0%: 0.069
   75.0%: 0.541
   90.0%: 0.750
   97.5%: 0.930

None

In R, we have a plot() that produces a traceplot of model terms like the global error scale \(\sigma^2\) or (if \(\sigma^2\) is not sampled) the first observation of cached train set predictions.

In Python, we provide a plot_parameter_trace() function for requesting a traceplot of a specific model parameter.

plot(bcf_model)

ax = plot_parameter_trace(bcf_model, term="global_error_scale")
plt.show()

For finer-grained control over which parameters to plot, we can also use the extractParameter() function in R or the extract_parameter() method in Python to query the posterior distribution of any valid model term (e.g., global error scale \(\sigma^2\), prognostic forest leaf scale \(\sigma^2_{\mu}\), CATE forest leaf scale \(\sigma^2_{\tau}\), adaptive coding parameters \(b_0\) and \(b_1\) for binary treatment, in-sample mean function predictions y_hat_train, in-sample CATE function predictions tau_hat_train) and then plot any subset or transformation of these values.

adaptive_coding_samples <- extractParameter(bcf_model, "adaptive_coding")
plot(
  adaptive_coding_samples[1, ],
  type = "l",
  main = "Adaptive Coding Parameter Traceplot",
  xlab = "Index",
  ylab = "Parameter Values",
  ylim = range(adaptive_coding_samples),
  col = "blue"
)
lines(adaptive_coding_samples[2, ], col = "orange")
legend(
  "topright",
  legend = c("Control", "Treated"),
  lty = 1,
  col = c("blue", "orange")
)

adaptive_coding_samples = bcf_model.extract_parameter("adaptive_coding")
fig, ax = plt.subplots()
ax.plot(adaptive_coding_samples[0, :], color="blue", label="Control")
ax.plot(adaptive_coding_samples[1, :], color="orange", label="Treated")
ax.set_title("Adaptive Coding Parameter Traceplot")
ax.set_xlabel("Index")
ax.set_ylabel("Parameter Values")
ax.legend(loc="upper right")
plt.show()